Dynamic system is a system whose states change with reference to an independent variable such as time.
- Moving car, states are the position, velocity, heading direction.
- Electric power generator, states are terminal voltage, frequency of the voltage wave form.
- Air conditioning system, states are the temperature, humidity in the indoor system
If the states are not controlled, they may end up at anywhere, this can be highly unacceptable and sometimes may lead to hazardous situation.
Mathematical model of dynamic system essentially are differential equations.
- Electrical circuits
Hence Kirchhoff voltage equation yield differential equations.
- Mechanical systems
Fs(t) = kx(t)
- Thermal systems
Covered by laws of thermodynamics recall newtons law of cooling for instance
Dynamic systems we consider are electrical, electromagnetic, mechanical, electromechanical, chemical and thermal or variety of combinations of them.
Laplace Transform for mathematical modeling
Laplace transform is a mathematical tool that can use to solve linear differential equations. The Laplace transform of a time domain function or signal f(t) is defined as
M = mass
FE = Engines? Breaking torque
D = damping constant
V = velocity
F = ma (Assume zero initial conditions)
GP(S) is call the “transfer function”
FE(t) Car V(t) converts to FE(S) Car V(S)
However, if we interest in position, u(t) of the car
By taking Laplace transform and assuming zero initial condition, X(0) =0
V(S) – SX(S) = 0
From equation (1)
If we want to understand how the current, i(t) in the circuit may be control by adjusting the supply voltage V(t) as the input, by using KVL
By taking Laplace transform and assuming zero initial conditions i(0) = 0
I(S) – Laplace input functions
E(S) is Laplace output functions
Control System in Laplace Domain
E(S) = R(S) – H(S)C(S)
GC(S)GP(S)E(S) = C(S)
m(S) is closed loop transfer function
Gc(S) is controller transfer function
Gp(S) is plant transfer function
H(S) is feedback element transfer function
C(S) is output transfer function
R(S) is reference input transfer function
Controller design explained
e(t) ——-> 0; E(S) ——-> 0
E(S) = R(S) – C(S)H(S) ——-> R(S) = C(S)H(S)
Stability of a control system
Stabilizing the control system must do prior to controller design and tuning. It is impossible to work with an unstable system. For instance, when you are learning to ride a bicycle, first step is to learn to stay stable (without falling or getting out of the road). Learning fine skills come later. Stability refers to firmness or steadiness, once the system has been stabilized optimizing the control performance is then possible.
Routh’s Stability criterion for mathematical modeling
If the transfer function of the system known, stability of linear systems can check using Routh’s criterion. Transfer function defined using familiar notation as
1 + Gc(S)Gp(S)H(S) = 0
This is usually a polynomial function of S that may written in the following form
Now, Construct the Routh’s array as follow
Condition for stability for mathematical modeling
Provided > 0, all the elements in the first column must be non negative.
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